An aircraft executes a hrorizontal loop of radius 1 km with a steady speed of `900 km h^(-1)` Compare its centripetal acceleration with the acceleration due to gravity ?

Explanation

To solve the problem, we need to find the centripetal acceleration of the aircraft executing a horizontal loop and then compare it with the acceleration due to gravity.

### Step 1: Understand the Formula for Centripetal Acceleration
The formula for centripetal acceleration (Ac) is given by:
\[ A_c = \frac{v^2}{r} \]
where:
- \( v \) is the speed of the object,
- \( r \) is the radius of the circular path.

### Step 2: Convert the Speed from km/h to m/s
The speed of the aircraft is given as \( 900 \, \text{km/h} \). We need to convert this to meters per second (m/s):
\[ v = 900 \, \text{km/h} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} = 250 \, \text{m/s} \]

### Step 3: Convert the Radius from km to m
The radius of the loop is given as \( 1 \, \text{km} \). We convert this to meters:
\[ r = 1 \, \text{km} = 1000 \, \text{m} \]

### Step 4: Calculate the Centripetal Acceleration
Now we can substitute the values of \( v \) and \( r \) into the centripetal acceleration formula:
\[ A_c = \frac{v^2}{r} = \frac{(250 \, \text{m/s})^2}{1000 \, \text{m}} = \frac{62500 \, \text{m}^2/\text{s}^2}{1000 \, \text{m}} = 62.5 \, \text{m/s}^2 \]

### Step 5: Compare with the Acceleration Due to Gravity
The acceleration due to gravity (g) is approximately:
\[ g = 9.8 \, \text{m/s}^2 \]
Now, we can compare the centripetal acceleration with gravity:
\[ \frac{A_c}{g} = \frac{62.5 \, \text{m/s}^2}{9.8 \, \text{m/s}^2} \approx 6.4 \]

### Conclusion
The centripetal acceleration of the aircraft is approximately \( 6.4 \) times greater than the acceleration due to gravity.

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